§ Simply connected spaces

• A space is simply connected iff fundamental group at all points is trivial.
• We usually don't want to talk about basepoint, so we assume that the space is path-connected. This means we can move the basepoint around, or not take about the basepoint.
• So, a path-connected space is simply connected iff the fundamental group is trivial.

§ Simply connected => all paths between two points are homotopic.

If $x, y$ are two points, then there is a single unique homotopy class of points from $x$ to $y$. Consider two paths from $x$ to $y$ called $\alpha, \beta$. Since $\beta^{-1} \circ \alpha \in \pi_1(x, x) = 1$, we have that $\beta^{-1} \circ \alpha \simeq \epsilon_x$. [ie, path is homotopic to trivial path ]. compose by $\beta$ on the left: This becomes $\alpha \simeq \beta$.

• This is pretty cool to be, because it shows that a simply connected space is forced to be path connected. Moreover, we can imagine a simply connected space as one we can "continuously crush into a single point".